I propose to present our recent results [LICS'19,LICS'21] on capturing various bisimilarity notions— covering not only the conventional relational notion but also its quantitative extensions such as probabilistic bisimulation and bisimulation metric—in the language of category theory. The theory combines abstract category theory and concrete games—so-called codensity games which are played in categories—which I believe will be of interest of Moshe and the audience.